A196810 Number of ways to place 2 nonattacking nightriders on an n X n cylindrical board.
0, 4, 18, 80, 200, 420, 756, 1472, 2358, 3860, 5500, 8304, 11232, 15484, 21090, 27392, 34816, 44604, 55404, 69840, 84294, 102124, 122452, 147264, 173800, 203476, 237762, 276752, 318304, 368340, 418500, 478208, 541398, 611524, 689780, 774576, 863136, 963148
Offset: 1
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- S. Chaiken, C. R. H. Hanusa and T. Zaslavsky, A q-queens problem I. General theory, arXiv:1303.1879 [math.CO], 2013-2014. See also. [_N. J. A. Sloane_, Feb 16 2013]
- V. Kotesovec, Non-attacking chess pieces
Programs
-
Mathematica
Table[(143*n^2)/30-(79*n^3)/15+n^4/2+16/5*n^2*Floor[n/5]+n^2*Floor[n/4]+4/3*n^2*Floor[n/3]+(n+2*n^2)*Floor[n/2]+8/5*n^2*Floor[(1+n)/5]+n^2*Floor[(1+n)/4]+2/3*n^2*Floor[(1+n)/3]+8/5*n^2*Floor[(2+n)/5]+8/5*n^2*Floor[(3+n)/5],{n,1,100}]
Formula
G.f.: -(2*x^2*(2 + 17*x + 96*x^2 + 384*x^3 + 1203*x^4 + 3100*x^5 + 6917*x^6 + 13670*x^7 + 24466*x^8 + 39974*x^9 + 60206*x^10 + 83709*x^11 + 107667*x^12 + 128088*x^13 + 141070*x^14 + 143882*x^15 + 136037*x^16 + 119239*x^17 + 96892*x^18 + 72808*x^19 + 50428*x^20 + 31926*x^21 + 18321*x^22 + 9388*x^23 + 4223*x^24 + 1622*x^25 + 514*x^26 + 127*x^27 + 22*x^28 + 2*x^29))/((-1+x)^5*(1+x)^3*(1+x^2)^3*(1+x+x^2)^3*(1+x+x^2+x^3+x^4)^3).
Recurrence: a(n) = a(n-32) + 4*a(n-31) + 10*a(n-30) + 17*a(n-29) + 20*a(n-28) + 11*a(n-27) - 15*a(n-26) - 54*a(n-25) - 90*a(n-24) - 99*a(n-23) - 63*a(n-22) + 18*a(n-21) + 116*a(n-20) + 188*a(n-19) + 194*a(n-18) + 123*a(n-17) - 123*a(n-15) - 194*a(n-14) - 188*a(n-13) - 116*a(n-12) - 18*a(n-11) + 63*a(n-10) + 99*a(n-9) + 90*a(n-8) + 54*a(n-7) + 15*a(n-6) - 11*a(n-5) - 20*a(n-4) - 17*a(n-3) - 10*a(n-2) - 4*a(n-1).
Explicit formula: a(n) = -n/4+(572*n^2)/225-(3*n^3)/2+n^4/2+(-1)^n*(n/4+n^2/2)+1/2*n^2*cos((n*Pi)/2)+16/25*n^2*cos((4*n*Pi)/5)+4/9*n^2*cos((4*n*Pi)/3)+16/25*n^2*cos((8*n*Pi)/5).
Chaiken et al. give a 4th degree quasi-polynomial formula. - N. J. A. Sloane, Feb 16 2013
Note that cited formula is for normal chessboard (not cylindrical), see sequence A172141. - Vaclav Kotesovec, Dec 09 2013
Comments